As it happens, the name schwi is helpful in naming a higher type of schwa, closer in sound to short I. You see, there is a hypothesis that English has two distinct types of schwas, one which is closer and one which is opener. The difference is epitomized by the phonetic contrast between ‘Georgia’s’ and ‘George’s,’ between ‘affect’ and ‘effect,’ or between ‘Lennon’ and ‘Lenin’ (in both these examples, the first one has the more open vowel). Among commentators on John Well’s English phonetics blog, the closer one is often called schwi and the opener one is often called schwa. (Wells himself has not embraced the term schwi.)
That’s the origin of the word schwi in a nutshell, but there’s more to it than that.